The fastest way to destroy Tank Destroyers is on Tank Superiority. Tanks will be everywhere, and you can destroy them with gadgets, Main Battle Tanks, or other Tank Destroyers.

As for the Mobile Artillery, you have to find one on Conquest or Rush. They can be found on every ConquestArmored Kill map, but a specific flag must be captured to make them available. The Mobile Artillery trucks are the BM-23 and M142. If you find one on Rush, it will be in the Attacker's base. Wait for the attackers to come with it, if they do drive, then destroy it with rocket launchers or other vehicles.